Severi-Bouligand tangents, Frenet frames and Riesz spaces

TL;DR

This paper links geometric tangent concepts with algebraic structures, showing that principal ideals in Riesz spaces relate to the existence of Frenet frames and Severi-Bouligand tangents in Euclidean spaces.

Contribution

It generalizes the connection between Severi-Bouligand tangents and Riesz space ideals from 2D to higher dimensions, introducing a method to compute Frenet frames from sample sequences.

Findings

Principal ideals in Riesz spaces characterize outgoing Severi-Bouligand tangents.

Frenet frames can be approximated from sample points without higher derivatives.

The results unify geometric and algebraic perspectives on tangent structures.

Abstract

It was recently proved that a compact set $X\subseteq \mathbb R^2$ has an outgoing Severi-Bouligand tangent vector $u\not=0$ at $x\in X$ iff some principal ideal of the Riesz space $\mathcal R(X)$ of piecewise linear functions on $X$ is not an intersection of maximal ideals. "Outgoing" means $X\cap [x,x+u]=\{x\}$. Suppose now $X\subseteq \mathbb{R}^n$ and some principal ideal of $\mathcal R(X)$ is not an intersection of maximal ideals. We prove that this is equivalent to saying that $X$ contains a sequence $\{x_i\}$ whose Frenet $k$-frame $(u_1,\ldots,u_k)$ is an outgoing Severi-Bouligand tangent of $X$. When the $\{x_i\}$ are taken as sample points of a smooth curve $\gamma,$ the Frenet $k$-frames of $\{x_i\}$ and of $\gamma$ coincide. The computation of Frenet frames via sample sequences does not require the knowledge of any higher-order derivative of $\gamma$.